Cut mesh in half with filled faces?
Previous answers will work, but two other options need to be given.
1. Booltool addon - add a Solidify modifier to the Plane object, and then shift-select your target object and choose Difference bool brush. 
2. With all the faces selected, you can use Shift-K to draw the knife tool through the mesh in edit mode so that it creates a cut even through odd ngons. After, you can select this new loop and use V to rip and then select one side and separate like explained before.